Section 1.5Quadratic Equations

2

Definition of a Quadratic Equation

A quadratic equation in x is an equation that can be written

in the general form

0,

where a, b, and c are real numbers, with a 0. A quadratic

equation in x is a

ax bx x

lso called a second-degree polynomial

equation in x.

Solving Quadratic Equations

by Factoring

The Zero-Product Principle

If the product of two algebraic expressions

is zero, then at least one of the factors is

equal to zero.

If AB=0, then A=0 or B=0

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Solving a Quadratic Equation by Factoring

1. If necessary, rewrite the equation in the general

form ax +bx+c=0, moving all terms to one side,

thereby obtaining zero on the other side.

2. Factor completely.

3. Apply the zero-product principle, setting each factor

containing a variable equal to zero.

4. Solve the equations in step 3.

5. Check the solutions in the original equation

Example

2 5 6 0x x Solve by factoring:

Example

Solve by factoring:

2 3 40x x

Example

Solve by factoring:

22 11 15 0x x

Graphing CalculatorThe real solutions of a quadratic equation ax2+bx+c=0 correspond to the x-intercepts of the graph. The U shaped graph shown below has two x intercepts. When y=0, the value(s) of x will be the solution to the equation. Since y=0 these are called the zeros of the function.

Solving Polynomial Equations using the Graphing Calculator

Repeat this process for each x intercept.

By pressing 2nd Trace to get Calc, then the #2,you get the zeros. It will ask you for left and right bounds, and then a guess. For left and right bounds move the blinking cursor (using the arrow keys-cursor keys) to the left and press enter. Then move the cursor to the right of the x intercept and press enter. Press enter when asked to guess. Then you get the zeros or solution.

Solving Quadratic Equations

by the Square Root Property

2

2

2

The Square Root Property

If u is an algebraic expression and d is a

nonzero real number, then u =d has

exactly two solutions.

If u , then u= d or u=- d.

Equivalently,

If u , then u= d.

d

d

Example

Solve by the square root property:

24 44 0x

Example

Solve by the square root property:

22 7x

Completing the Square

22

22

2

Completing the Square

bIf x bx is a binomial, then by adding ,

2

which is the square of half the coefficient of x,

a perfect square trinomial will result. That is,

b bx bx+

2 2

x +8x

x

2

22

8 add

2

-7x 7x add

2

Why we call this completing the square.

Example

What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.

2

2

10

9

x x

x x

Example

Solve by Completing the Square:

2 8 10 0x x

Example

Solve by Completing the Square:

2 14 29 0x x

Solving Quadratic Equations

Using the Quadratic Formula

2

2

The Quadratic Formula

The solutions of a quadratic equation in general

form ax bx+c=0, with a 0, are given by the

quadratic formula

-b b 4x=

2

ac

a

Example

Solve by using the Quadratic Formula:

2

2

6 30 0

2 5 8 0

x x

x x

The Discriminant

Example

Compute the discriminant and determine the number and type of solutions:

2

2

2

2 3 7 0

5 4 0

6 1 0

x x

x x

x x

Determining Which

Method to Use

Applications

The Pythagorean Theorem

The sum of the squares of the

lengths of the legs of a right

triangle equals the square of the

length of the hypotenuse.

If the legs have lengths a and b,

and the hypotenuse has len2 2 2

gth c,

then a b c

Example

A machine produces open boxes using square sheets of metal. The

figure illustrates that the machine cuts equal sized squares measuring

2 inches on a side from the corners, and then shapes the metal into an

open box. Write the equation for the volume of this box. If the volume is

50 cubic inches, what is the length of the side of the original metal.

Example

A 42 inch television is a television whose screen’s diagonal length is 42 inches. If a television’s screen height is 26 inches, find the width of the television screen

42 inches

26 inches

(a)

(b)

(c)

(d)

2

Solve by the square root property.

x-4 15

4 15

4 15

19

4 15

(a)

(b)

(c)

(d)

2

Solve by completing the square.

x 12 3 0x

4 39

6 33

6 33

12 39